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Can we apply Bernoulli equation to a closed system the same way we apply it to an open system? In an open system increasing the speed of the fluid implies that either the pressure or GPE should decrease as a result of conserving energy. I wounder how would this apply to a closed system?

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In a closed system where there is no flow and thus velocity = 0, Bernoulli's equation is reduced to $$ P_1 + \rho gz_1 = P_2 + \rho g z_2$$ $$ P_2= P1 - \rho g \Delta z$$ which is really just the normal rule for pressure in liquids. In gases usually $\rho$ is too small so the pressure is constant throughout the gas. So basically it is reduced to the normal rules you use for fluid statics.

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I suppose the OP thinks of a closed circuit with flow, provided by a pump.

This type of circuit exists for example in a PWR primary circuit, where liquid water is heated by the nuclear fuel in the reactor pressure vessel and continuously pumped to a heat exchanger, called steam generator, where a secundary (also closed) circuit delivers the steam to the turbines.

The Bernoulli equation is valid because it is a steady state flow. The presure at the small pipes of the heat exchanger is smaller than at the large entry chamber for example.

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